### Different number systems Representation of numbers in binary

```Number Systems
Different number systems
Representation of numbers in binary
Conversion between decimal and binary,
Use of subscripts 2, 10 and 16 for bases
Number Systems

Decimal number system – Base 10 = 1, 2
,3 4, 5, ect..

Binary number system –Base 2 = 0001,
0010, 0011, ect…

Hexadecimal number system = Base 16 =
9, A, B, 4C ect…
Decimal Number Systems

Decimal numbers are base 10
They are made up of 10 numbers –
0,1,2,3,4,5,6,7,8,9.
 Combining the ten numbers will create
units, tens, hundreds and thousands

Hundreds
Tens
Units
341
102
101
100
300
40
1
300 + 40 +1 = 341
Split the following decimal numbers
Hundreds
Tens
Units
550
Hundreds
Tens
982
Units
Hundreds
Tens
Units
550
102
101
100
500
50
0
500 + 50 + 0 = 55010
Hundreds
Tens
Units
982
102
101
100
900
80
2
900 + 80 + 2 = 98210
Binary Number System
Binary numbers are base 2
 Computer language
 They are made up of 2 numbers – 1 and 0

Decimal
Binary
Decimal
Binary
010
02
510
1012
110
12
610
1102
210
102
710
1112
310
112
810
10002
410
1002
910
10012
 Computer memory locations
 They are made up of 16 numbers

Decimal
Hex
Decimal
Hex
Decimal
Hex
Decimal
Hex
010
016
510
516
1010
A16
1510
F16
110
116
610
616
1110
B16
210
216
710
716
1210
C16
310
316
810
816
1310
D16
410
416
910
916
1410
E16
Importance of Base numbers

Writing the base numbers is very
important as;
◦ 1510 and 1516 are not the same number but
without the base they would be both
considered as the same number
◦ 1010 and 102 are not the same
number as 102 represents 210
Complete the table
Number
2010
2A16
10101012
10110
1516
1110001112
Number System
Number
Number System
2010
Decimal
2A16
10101012
Binary
10110
Decimal
1516
1110001112
Binary
Converting
Binary to Decimal
Explanation
1.
Write down the placement value on top
of each number.
24
23
22
21
20
16
8
4
2
1
2.
Write the values that are on (the ones
with a one under them
3.
Example

We want to convert 110012 to decimal
24
23
22
21
20
1
1
0
0
1
16
8
4
2
1
16
8
1
16 + 8 + 1
25
Working

1.
2.
3.
4.
5.
Convert the following to decimal
1010102
1110112
101010012
0011001112
1110101002

1.
2.
3.
4.
5.
Convert the following to decimal
1010102 = 4210
1110112 = 5910
101010012 = 16910
0011001112 = 10310
1110101002 = 46810
Converting
Decimal to Binary
Method One
1.
Write down the placement values of
binary
124 64
32
16
8
4
2
1
2.
Chose the numbers that add up to you
decimal number
3.
Put a 1 under the numbers used to add
Example

Convert 4610 to binary
124 64
0
0
32
16
8
4
2
1
1
0
1
1
1
0
32 + 8 + 4 + 2 = 46
4610 = 001011102
Method Two

Divide the original number by 2 and write
down the remainder even if it is 0

Keep on dividing the decimal numbers by
2 until 1 is divided by 2

Write down the remainders next to each
other starting from the bottom moving
upwards
Example

Convert 4610 to binary
46 /
2
=
23 r
0
23 /
2
=
11 r
1
11 /
2
=
5
r
1
5
/
2
=
2
r
1
2
/
2
=
1
r
0
1
/
2
=
0
r
1
Ans  4610 = 1011102
Working
Convert the following decimal numbers
to binary
1. 1010
2. 6610
3. 12010
4. 3510
5. 8810

Convert the following decimal numbers
to binary
1. 1010 = 10102
2. 6610 = 10000102
3. 12010 = 11110002
4. 3510 = 1000112
5. 8810 = 10110002

Converting
Explanation
Split the binary number into groups of 4
1001110 = 0100 – 1110
 Write the 2x on top of each number
starting from the right
 Add the numbers that are on
 Write down the totals, if a total is larger
than 9, convert it 0 1 0 0 1 1 1 0
to the hex letter 23 22 21 20 23 22 21 20

NOTE: when we do not
have enough bits lefts to
create a group of 4 we add
0s
8 4 2
4
1
8 4 2 1
14
4E16
Example

0
1
1
0
0
1
1
1
23
22
21
20
23
22
21
20
8
4
2
1
8
4
2
1
6
7
6716
Working

1.
2.
3.
4.
5.
1110101002
11101112
1010102
1112
11100012
Working

1.
2.
3.
4.
5.
1110101002 = 1D416
11101112 = 7716
1010102 = 2A16
1112 = 716
11100012 = 7116
Converting
Explanation
1.
2.
3.
Write each individual number in the
Write the binary placement values for
each hex number
List 1s under the placement values that
are on
B = 11
4
4. Write
the split 23 22 21 20 23 22 21 20
binary number as 8 4 2 1
8 4 2 1
one whole
1 0 1 1
0 1 0 0
number
101101002
Example

Convert 2C16 to binary
2
C = 12
23
22
21 20
23 22
21
20
8
4
2
8
2
1
0
0
1
1
0
1
001011002
4
1
0
0
Working

1.
2.
3.
4.
5.
Convert the following hex numbers to
binary
AB16
F716
1516
CC16
2216

1.
2.
3.
4.
5.
Convert the following hex numbers to
binary
AB16 = 101010112
F716 = 111101112
1516 = 000101012
CC16 = 110011002
2216 = 001000102
Converting
Method One
Divide the decimal number by 16 taking
note of the remainders
 Keep on dividing the whole number by 16
until the whole number obtained is 0.
 Write down the remainders next to each
other starting from the bottom, changing
numbers greater than 9 to letters

465 /
16
=
29
r
1
29
/
16
=
1
r
13
1
/
16
=
0
r
1
ANS = 1D116
Example

800 /
16
=
50
r
0
50
/
16
=
3
r
2
3
/
16
=
0
r
3
ANS = 32016
Method Two
Convert the decimal number to binary
2. Convert the binary number to
1.
Example

512 256 128 64
1
1
0
0
32
16
8
4
2
1
1
0
0
0
0
0
512 + 256 + 32 = 800
80010 = 11001000002
0 0 1 1
23 22 21 20
8 4 2 1
3
0 0 1 0
23 22 21 20
8 4 2 1
2
32016
0
23
8
0 0 0
22 21 20
4 2 1
0
Working
numbers
1. 34010
2. 11910
3. 6610
4. 2510
5. 11110

numbers
1. 34010 = 15416
2. 11910 = 7716
3. 6610 = 4216
4. 2510 = 1916
5. 11110 = 6F16

Converting
Explanation
Writing down the placement values on
top of each number starting with 160
 Multiply the top value with the

Converting
43A16 to
decimal
162
161
160
256
16
1
4
3
A
(256x4)
(16x3)
(1x10)
1024
48
10
=1024+48+10
=108210
Working

1.
2.
3.
4.
5.
Convert the following into decimal
5516
CB16
F816
B416
9016

1.
2.
3.
4.
5.
Convert the following into decimal
5516 = 8510
B016 = 17610
2F816 = 76010
B416 = 18010
9016 = 14410
Homework

Copy and complete this table
Decimal
2110
Binary